Content-Type: text/shitpost


Subject: Non-transitive relations
Path: you​!your-host​!walldrug​!prime-radiant​!berserker​!plovergw​!shitpost​!mjd
Date: 2018-05-02T18:12:47
Newsgroup: talk.mjd.math.non-transitive-relations
Message-ID: <64b0f1e9b2f83316@shitpost.plover.com>
Content-Type: text/shitpost

The phases of an eight-phase dual-ring intersection provide an interesting example of a non-transitive relation.

This is a diagram of the
intersection of two two-way streets, as seen from above.  Each of the
four incoming roads has an arrow showing the direction of through
traffic and another showing the direction of left-turning traffic.
Clockwise in order the through-traffic arrows are labeled with
Φ2,Φ4,Φ6,Φ8, and the corresponding left-turn arrows are labeled
Φ5,Φ7,Φ1,Φ3.  Arrows Φ5 and Φ1 are accompanied by green traffic
signals, and Φ4 and Φ6 by red signals.

Let !!a!! and !!b!! be phases, and write !!a\sim b!! when the phases are compatible, which means that traffic making those movements will not collide. This relation is reflexive and symmetric, but not transitive, because for example we have !!\Phi6\sim \Phi1!! and !!\Phi1\sim \Phi5!! but not !!\Phi6\sim\Phi 5!!. A mathematician would have numbered the phases differently, but even with the standard numbering the rule is not too complicated:

$$ a\sim b \text{ when any of these holds:} \begin{cases} a\in\{\Phi1, \Phi 2\}\text{ and }b\in\{\Phi5, \Phi6\},\text{ or} \\ a\in\{\Phi3, \Phi 4\}\text{ and }b\in\{\Phi7, \Phi8\},\text{ or} \\ a = b \end{cases} $$